Would it be a problem if they did? And, as far as I know, GLAAD (formerly the Gay & Lesbian Alliance Against Defamation) doesn’t sponsor LGBT pride events either, as far as I know. That’s usually left to LGBT people in each city where these events take place.
I’m not sure what the problem is with using the word “pride”. It has many different shades of meaning depending on the context. I doubt that most people would think it bad if someone said on the 4th of July that they’re “proud to be an American”, for example. That kind of pride is not really the same kind which is considered to be one of the “seven deadly sins”.
Yes, it would be a problem. Anti-defamation for the C-L means defending the Church against unjust attacks. It doesn’t mean promoting a lifestyle. Promoting Catholicism is a job for evangelization and apologetics.
Are all these different communities pushing an “agenda” because they have these kinds of organizations, or is it only the LGBT community that has an “agenda”?
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